Note that the “Tile \(\Rightarrow \)  Spectral” direction follows from [1]}. Hence, we only need to prove the “Spectral \(\Rightarrow \)  Tile” direction.
When we consider Fuglede's conjecture in cyclic groups, one of the most important tools is the so-called (T1) and (T2) conditions, which was introduced by Coven and Meyerowitz [2]}. In this paper, we introduce a new tool “group ring” to study spectral sets in cyclic groups. In particular, we prove that Fuglede's conjecture holds in \(\mathbb {Z}_{p^{n}qr}\) . This paper is organized as follows. In Section , we recall some basics of spectral sets and tiles in cyclic groups. In Section , we prove some useful lemmas by group ring. In Section , we prove Theorem REF .
